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I have just now two new master students who are going to look into certain geometric aspects of physics. Also a colleague just asked me for suggestions for a course on “geometry and physics”. I kept pointing to Frankel’s book. That’s great as far as it goes, but it misses on a lot of clarifications available meanwhile.
So I thought it’s about time that I start making notes for a modern introductory course on
I put that into the $n$Lab proper, instead of on my personal web. One reason is that otherwise hyperlinking becomes a pain. Another reason is that this should really not be hidden and reserved somewhere but be out there in the open for everyone to join in. Though I do have a certain strategy in mind, which I would like to ask to follow.
You’ll see what I mean when you look at the entry. It’s so far just a first sketch of a section outline with some keywords and notes to indicate what is eventually to go there. That’s how far I got tonight. (And I really need to sleep now to be ready for my homological algebra course tomorrow…) But I guess the idea and the intended structure is already visible. Will be expanded and edited in the course of the next weeks.
Wow, I’d like to be able to sit in on such a course!
Thanks, Todd. Well, let’s see how it pans out. :-)
But I’ll be building up things incrementally here and would be most pleased about discussing things here step-by-step, and about whatever input or comments especially you may have!
I want to write out the “first lecture” this afternoon. Right now I really need to run. But this morning it occurred to me that I’ll be doing the writeup of each “session” in three “layers”.
A colleague asked me today in which generality the concept of “momentum space” has (invariant) geometrical meaning in classical mechanics. There is no problem with coordinate space and with phase space (basically cotangent bundle). I said that momentum space has sense only if I can identify somehow the fibers of cotangent bundle, like I can do for the usual affine space.
It is very nice Urs is sharing his grand vision. I hope we can also gradually fill in some links and answers to classical questions in the entry, like this one.
Maybe it makes sense to link it to books and reviews in mathematical physics.
in which generality the concept of “momentum space” has (invariant) geometrical meaning in classical mechanics.
Given a symplectic manifold regarded as phase space, then for every choice of polarization coordinate space is the leaf space. So momentum space is, to the extent that this exists (and it may not exist) the quotient of the phase space by the leaf space. The typical leaf.
I hope we can also gradually fill in some links and answers to classical questions in the entry, like this one.
That’s the aim of geometry of physics, to provide lots of concrete information. I’ll not be deviating in this from the other books. I’ll just use a better toolbox.
I am familiar with polarizations in the context of quantization; different polarizations are there on equal footing as different quantizations exist. Now are there some common principles in choosing preferred polarization to agree with the “usual” momentum space in several most classical examples ? (I understand that the question is likely ill-defined but if there is something to say, some principle you know I’d like to hear).
P.S. Maybe my expectation that the choice of polarization for this purpose is any better question that how to choose it for the purposes of (pre)quantization is completely misleading.
The choice of polarization is not just relevant in quantization, but already in the classical Hamiltonian mechanics. The choice of polarization on a symplectic manifold is precisely a choice of “canonical coordinates” and “canonical momenta”. The “canonical coordinates” label the leaves of the polarization and the “canonical momenta” parameterize each leaf.
(This should be put into the entry, but I have no time right now…)
Then in quantization we say “a wavefunction is a function on phase space that depends just on the canonical coordinates”. Hence that it is is (covariantly) constant along the leaves, namely along the “canonical momentum” direction.
This is a great answer. I should have known this from the study of Legendre transform (which leads to usual definition of generalized momenta in physics), but did not look at it in this picture.
Now I have some time. I added some paragraphs along the above lines to polarization and also to canonical momenta. I have also made canonical coordinates redirect to polarization.
I have now written a first actual subsection, under
This is meant to be the pre- or meta-section where we say a word about the basic geometric object on which differential geometry is modeled, namely the real line, and how it is related to “geometry of physics”.
Suggestions, comments, etc. are most welcome.
As well as the speculative alternatives you mention, such as the p-adics, what of supergeometry?
I mean would you need to change
The abstract worldline of any particle is modeled by the continuum real line R,
working natively in supergeometry?
Yes, I’ll do that in the section Supergeometric coordinate systems later.
The abstract worldline of a fermionic particle is a $\mathbb{Z}_2$graded formal neighbourhood $\mathbb{R}^{1|n}$ of the real line.
I have been further working on the “first session”. It still needs some adjustments and a bit more stuff sprinkled in here and there, but it’s beginning to walk like the duck that it’s supposed to be.
The subsection layout of the first session is now this:
Please comment etc. And: this is not “my page”. You can go there and edit right away, if you feel like it, as usual. I guess by now it is clear what kind of structure this entry is meant to have. Let’s just discuss changes here, as usual.
I was inspired by this to pick up Frankel’s book. I absolutely love what I’ve looked at. I double majored in math and physics as an undergrad, then went on to math. Now that I have a much more firm grasp of differential geometry and advanced mathematical topics I always wondered how these physics notions could be placed in them.
I am now writing the “second session” of geometry of physics, titled Smooth Spaces.
In its first “layer” Smooth Spaces – Layer Mod my aim is to explain sheaves on $CartSp$ in an intuitively transparent (but nevertheless precise) way without referring to any actual sheaf theory at this point (that is instead the topic of Smooth Spaces – Layer Sem which readers may want to ignore).
So I have what should be easily absorbable sections
Ideal would be a layperson willing to have a look at that and willing to give me feedback.
Difficult to tell who would qualify as a layperson here. I am however willing to give some feedback, when I have a suitable moment to look at it.
Thanks, Todd. Whatever kind of comment you might have, I’d like to hear it. But please keep in mind that it’s not meant to be finished yet.
Okay, no worries Urs.
I did take a look just now, and what particularly caught my eye was “on the horizon: path integral”. Does this mean that you have some ideas about putting this on a rigorous footing (in quantum field theory)? That would be pretty exciting, and I’ve long wondered whether and how smooth differential geometry might be put to service here.
There’s not planned anything here about the path integral as a quantization procdure. Instead, I’ll discuss quantization in the form of geometric quantization.
But all of the aspects of “setting up the path integral” will be discussed, namely smooth spaces “of paths” (histories) with smooth action functionals on them. And the corresponding “path integral quantum anomalies” will be discussed, namely the cases where these action functionals are in fact not smooth functions but are sections of smooth circle bundles over these spaces histories.
At the point where it currently says “on the horizon: path integral”, which is very much at the beginning of the text, I just want to briefly remind the physics-inclined reader why considering mapping spaces at all is something he will be interested in.
quantization in the form of geometric quantization
Are you saying that there are rigorous results in geometric quantization for field theory as well, or we are talking here just finite dimensional mechanics (sorry if I missed a possible answer up in the thread).
For Chern-Simons-type quantum field theories geoemtric quantization is the method of choice. This then also gives all their holograohic dual theories.
But more generally, the entry is of course aiming for the higher geometric quantization of multisymplectic geometry, which is meant to be designed to handle the case of field theory.
Of course there remain gaps in the story as found in the literature. But that’s good because, after all, the two students for whom I am doing this writeup at the end want an open question left to work on… :-)
What makes the “multisymplectic” to easier deal with the typical infinite-dimensionality problems arising when generalizing to field theory ?
The multisymplectic geometry is supposed to be the de-transgression of the traditional story, and hence the “extended phase space” is finite dimensional:
For instance
where the phase space of a 2-dimensional $\sigma$-model QFT is an infinite-dimensional loop space/path space $[\Sigma_1, X]$, carrying a prequantum 1-bundle….
…the corresponding extended phase space in multisymplectic geometry is the finite dimensional manifold $\Sigma_1 \times X$ (regarded as a “field bundle” $\Sigma_1 \times X \to \Sigma_1$), but carrying a prequantum 2-bundle.
The ordinary phase space is recovered as the relative mapping space / space of sections of the finite dimensional extended phase space and the ordinary prequantum bundle is supposed to be the transgression of the prequantum 2-bundle to this mapping space.
This is great (very helpful) answer Urs!
Can this philosophy of reducing to finite-dimensional case apply to the perturbatively defined field theories (i.e. for fairly general perturbations), or just to special “topological” examples ?
So this is not restricted to topological QFT. Some references on non-toplogical 2d QFT in multisymplectic refinement are at String - References - In multisymplectic geometry.
But currently the number of articles in the literature that genuinely discuss geometric quantization in multisymplectic geometry is tiny. See at Multisymplectic geometry - References - Quantization.
But with a little luck, in a year from now there will be two new nice theses improving on this situation… :-)
You are so amazing in spotting the most important developments early! I considered multisymplectic geometry as an exotic extension for some specific purposes and did not see its central need.
I have been working on the “3rd session”
There’d be more to do, but that’s all for which I seem to have the time at the moment. And at least it should be a fairly coherent story now.
I am reading – and enjoying very much! – this page. I spent the last hour or so looking through it and correcting typos. For the most part it reads pretty smoothly.
Thanks, Todd! For the feedback and for fixing typos. I am hoping to find more time for this in two weeks from now.
It turned out that my original plan for the “Syn-Layer” of this section didn’t quite work out. First I thought, as you can see, that it would be a good idea to combine the discussion of differential forms with that of concretifications of objects in a local topos. For that mechanism will play a central role when later we derive canonically existing action functionals in the cohesive context.
But when I finally found time to work on the Syn-Layer of “Differential forms” I had to realize that for discussing images in type theory as necessary for concretification, I really should have introduced identity types and bracket types before (of course).
Maybe I should leave it the way it is now but add a pointer that the reader should really come back to this only after reading some later sections. Or I reorganize it all. Haven’t decided yet.
I have started writing out some basics at
Somebody asked me by email about how exactly ordinary differential forms are recovered in cohesive homotopy type theory. I have started adding some pointers for this at
But this is still very skeletal as far as the text goes and not meant to be read now unless you want to be reminded of these pointers right now.
I have been working on the section that used to be titled “Tabulated index”, but which has grown now and which I have renamed for the moment to
The idea of this section is to go in natural language through basic constructions that one may want to talk about in physics, translated step-by-step to formal syntax and finally list what structures that gives rise to under categorical semantics, with pointers to the chapters further below that discuss this in detail.
At the same time I am trying to indicate how the required flavor of type theory is built up layerwise from the requirements of which kinds of statements we want to be able to make in physics.
I am fond (at the moment) of how it starts out, but somewhere around the middle you see me run out of steam with more ellipses appearing and less streamlined text. I’ll continue to work on it tomorrow.
This here is in the hope that unfinished as it is, readers can see what it’s aiming for and maybe already make comments.
Only a triviality, but
One says we need a notion of how points cohese together.
The verb is cohere, although there are signs of a colloquial shift. But then you’d also need to change ’DeCohese’ to ’DeCohere’ and maybe the worry then is that you’d be competing with decoherence.
Then what to do with
there is the the decohesed homotopy type ?
The verb is cohere
Can we think about that for a sec, David? There is a pronounced difference between the notions expressed by the nouns
coherence
cohesion
Similarly the corresponding verbs should be different. Now I gather that while it is standard to associate a verb with “coherence” traditionally one does not use a verb for “cohesion” so much.
But given the state of affairs, and in particular in a mathematical context where we are allowed to invent technical terms to a certain degree, there should be such a verb. But it should in any case not be the same verb as for coherence. For “exhibiting coherence” is a notion rather unrelated to “exhibiting cohesion”.
Wouldn’t you agree? Let me know if you think I am mixed up.
I have further worked on The full story in a few formal words. Now it should be getting both more coherent and more cohesive :-)
I had the thought of closing a full circle, exposition-wise, and arriving at a punchline to leave the reader in a state of curiosity, by
beginning the section with suggesting that to speak about physics we need to be able to judge that “There is…” and then using that to lead over to judgements in type theory
closing the section by indicating that after adding homotopy cohesion to the language we can say “There is light.” (namely $\vdash \; \nabla_{em} \colon [X, \mathbf{B}U(1)_{conn}]$) and then I added …as in “Let there be light.”
But maybe that last bit I should remove again. Not sure.
Re #35, I wasn’t arguing against neologisms, just making sure you were aware you were doing so.
Now I gather that while it is standard to associate a verb with “coherence” traditionally one does not use a verb for “cohesion” so much.
Well, the cohere-cohesive link is pretty strong in people’s minds, as also with adhere-adhesive for glue.
“exhibiting coherence” is a notion rather unrelated to “exhibiting cohesion”.
I see them as throughly related. A coherent argument is one in which the parts stick together in a good way.
Let’s see. We have
cohere (v.) Look up cohere at Dictionary.com 1590s, from L. cohaerere “to cleave together,” in transferred use, “be coherent or consistent,” from com- “together” (see co-) + haerere “to stick” (see hesitation). Related: Cohered; cohering.
(Cleave is to be avoided as it has two contradictory meanings, and was the subject of a forgettable discussion at the Cafe.)
I’d never made the connection with hesitation before.
c.1400, from O.Fr. hesitacion or directly from L. haesitationem (nom. haesitatio) “a hesitation, stammering,” figuratively “irresolution, uncertainty,” from haesitare “stick fast, remain fixed; stammer in speech,” figuratively “hesitate, be irresolute, be at a loss, be undecided,” frequentative of haerere “stick, cling,” from PIE *ghais-e (cf. Lith. gaistu “to delay, tarry”), from root *ghais- “to adhere; hesitate.”
Hmm, I’m not sure what to suggest. I can see you’d like a new verb, but I can make a firm prediction that wherever you present ’cohese’, people will be attributing this to your not being a native English speaker.
Re #36, I can see the amusement value adding “Let there be light” to the story. On a (slightly more) serious note, the phrase summons up for me the history of the debates as to the immanence or transcendence of God in the universe. A God such as Spinoza’s who is purely immanent, one aspect of the whole, another of whose faces is Nature, is not a God to bring light into existence by decree.
Then I’m led to reflect on the many kinds of speech acts (Austin, Searle, etc.). Your talk of judgements would seem to correspond to the use of language as descriptive. Austin noticed that speech performs other tasks, making promises, declaring two people married or a session of parliament open. The latter is particularly interesting when the speech act brings about a new state of affairs, and seems closest to “fiat lux”.
people will be attributing this to your not being a native English speaker.
I thought of this also. I too am not against neologisms (nor am I against a single word like ’decohere’ having more than one meaning), but if ’decohese’ is what’s put down, I think a footnote would be in order, to assure people it’s quite intentional and not a mistake.
I see, thanks for the dictionary pointers. I was with Jim here in that I don’t think of coherence as being unlike cohesion.
But if it sounds bad, I need to change it. However, I don’t want to change it to “decohere” because in a context of quantum mechanics “decoherence” in turn really means something entirely different.
So I need another term altogether. Hmm…
I am now taking the easy way out, which may actually have its merits anyway: I’ll abbreviate to $DeCoh$. And similarly then I’ll abbreviate $UnderlyingBundle$, which is too long anyway, to $UnderlBund$. Etc.
UnderlBund
Almost looks German to me! Austrian German, perhaps. :-)
True, now that you say it… Makes me want to change it back again… :-)
Unfortunately, I have to say that I think “cohese” is one of the ugliest neologisms I’ve ever seen. I doubt if I would ever be able to bring myself to use it. (-: Especially since the existent English word “cohere” has exactly the correct meaning, and I don’t think I’ve ever heard the verb form “cohere” used in reference to the category-theoretic meaning of “coherence”.
If the only worry is a clash with the quantum-mechanics meaning of “decoherence”, then why not choose another word that is an antonym of “cohere”? Like “dissolution” or “detaching”, maybe.
Maybe dissolve is good, because as opposed to “detach” that carries with it the idea that the result still has cohesion, but in a very unconstrained way – which is what $\sharp$ produces. Detaching is more what $\flat$ does.
But I’ll stick with $DeCoh$ now, out of laziness. Or let’s say: since I have more important things to worry about.
I have been further trying to beautify
I am beginning to think it is ready for prime time.
Supposed I read this out to you (always columns 1+2 first, then selected pieces of 3+4, not the whole thing in full detail.) How unhappy would you be? Which piece would you be most unhappy about?
I’ll give this a go over the next few days. On a trivial note, I gave it a quick run through, and didn’t know whether the $[X,\mathbf{B}\left(U\left(1\right)\times U\left(2\right)\times U\left(3\right)\right]$ is just missing a bracket, or also some ’S’s. The disappearance of the $_conn$ is due to forgetting, but there’s presumably no reason for the $SU$ to become $U$:
Given a gauge field $\nabla$ as above, there is an underlying instanton sector, $UnderlyingBundle(\nabla)$, in the collection $[X,\mathbf{B}\left(U\left(1\right)\times U\left(2\right)\times U\left(3\right)\right]$ of instanton configurations in the standard model.
Thanks, David!!
That’s quite awesome that you find missing closing parentheses and “S”s even. I have fixed that now. Thanks.
The odd feeling I have reading the section is that it gives no suggestion that there’s anything problematic about our understanding of current physics. It allows us to speak of what there is, the states of things and how these change, but no concerns appear about the kind of issue we discussed here.
Now perhaps it’s not the job of the formalism to do this. Of course it’s a great thing to be able to address the quantization of constructions from differential geometry. But wouldn’t life be wonderful if the casting of modern physics in cohesive homotopy type theory were to shed some light on what is perplexing about fundamental physical concepts, space, time, cause, etc.?
Is there, for example, any scope to try to make sense of Liang Kong’s
…our physical space is nothing but a network of structured stacks of information, from which spacetime can emerge.
I should add towards the end a warning that currently pre-quantization is fully formalized. There is still one missing item: at the moment I still don't know how to properly axiomatize the choice of polarization.
But concerning that other discussion we had: this is addressed already in the very setup of the discussion right at the beginning: instead of providing just the axiomatization of the quantum field theory itself, we are here concerned with adding to that its origin in a process of quantization. That equips the abstract space of states that a cobordism representation assigns to a manifold with an explicit geometric interpretation of its elements. This information not only of the quantum theory but also of the action functional which it is induced from is what allows to interpret the cobordism representation physically.
I should try to add a paragraph on that. But I won't do it today. I am rather time pressured with preparing a seminar talk on presentable $\infty$-categories today.
The very existence of polarization seems difficult. Isn’t that condition something close to an expression of having a classical integrable system ? Any comments ?
The largest class of integrable systems which have good geometrical theory is the Hitchin system. Beilinson and Drinfeld wrote several hundred pages long treatise on the quantization of that system.
Isn’t that condition something close to an expression of having a classical integrable system ? Any comments ?
No, why? It’s the precondition for having any quantization at all.
Urs: I am talking (in the first paragraph) what it is before quantization (hence I do not understand “at all”). If you have a global Lagrangean foliation, this is much like separation in action-angle variables, isn’t it ?
Second, at least for Hitchin system, what is a substantial subset of known integrable systems, one has explicitly the description in terms of Lagrangean foliations (and in generalizations, of Lagrangean fibrations).
I do not understand why are you so negative about the insight of integrable systems, where things work better similar like TQFTs are easier to understand than general QFTs, hence their omnipresence in general nonsense works.
Oh, I think I had a view of quantization as something like a ’fix’. We have a classical description and by some kind of magic it allows us to arrive sometimes at a workable quantum theory, along with the thought maybe that the direct construction of a quantum theory would be better. So really quite a lot like categorification, i.e., a non-algorithmic process to take us to some better place, where unnecessarily imposed conditions no longer apply. But you’re presenting quantization in a positive light.
Zoran: why do you say that having canonical momenta is the same as having an integrable system?
Why do you say I am negative towards integrable systems? I am just very doubtful of your sugestion that the existence of polarizations amounts to having an integrable system. I mentioned geometric quantization because your suggestion would imply that all of geometric quantization is only about integrable systems. Which clearly it is not.
why do you say that having canonical momenta is the same as having an integrable system?
Classical integrable system means having global action-angle variables. This is one of the rather standard expressions of Liouville’s integrability, as in the book of Arnol’d.
because your suggestion would imply that all of geometric quantization is only about integrable systems. Which clearly it is not.
Yes, let’s find out where the difference lays, my suggestion was that they are alike, there must be some difference, and still if you have an integrable system the geometric quantization prescription must be easier (though other quantization procedures may be then more insightful). Maybe geometric quantization knows how to deal with some degeneracies or something…
@David:
yes. I have a little remark on that in the “Full story in a few formal words”:
while it is true that there are “fundamental” QFTs which do not seem to arise from quantization of an action functional (notably that 6d (2,0)-supersymmetric QFT) these are holographic duals of higher dimensional QFTs which do arise from quantization of an action functional. And holography is supposed to be part of the “extended geometric quantization” that is axiomatizable in cohesive HoTT (the basic example for this that we have is that the transgression of the fully extended Chern-Simons action functional to codimension 1 yields the holographic dual WZW-model prequantum bundle).
The fundamental physics of the observed world (notably the standard model of particle physics) is governed by what is called quantum theory.
With the possible exception of gravity?
@Zoran:
so let’s disentangle these notions:
a polarization is just any foliation of phase space by Langrangian submanifolds. There is not even necessarily a Hamiltonian specified at this point.
a Liouville integrable system on a 2$n$-dimensional phase space is one that admits a polarization which is generated from the flow of $n$ Hamiltonians, wich each Poisson-commute with each other.
So the second is a much stronger condition than the first. The first is supposed to alsways exist for physics to make any sense at all. The second is the definition of Liouville integrability.
The fundamental physics of the observed world (notably the standard model of particle physics) is governed by what is called quantum theory.
With the possible exception of gravity?
This seems to be most unlikely. It would break everything that is known about theoretical physics if gravity is fundamentally a classical field theory coupled to a stress-energy tensor of a matter QFT. The standard introductions to quantum gravity go at length through all the mess that results when trying to even just formulate this. It’s not a mathematical theorem that this can’t work, but it seems entirely unlikely.
And gravity is not the first non-renormalizable QFT that mankind ran into. Every single other non-renormalizable QFT that ever occurred ended up being an effective QFT of a more fundamental UV-completion that is renormalizable. Mankind would have been ill-advized to give up on all these and hypothesize that they have to be classical QFTs instead. Moreover, it is clear that whatver the UV-completion of gravity is, it is out of range of current experiments. So assuming that there is quantum gravity is the only theoretically plausible assumption and not in contradiction with any experiment. There is no good reason to give up this assumption and very good reason not to give it up.
Todd, I have expanded that first sentence at The full story in a few formal words as follows (should be optimized, but that’s what I have for the moment):
The fundamental physics of the observed world is governed by what is called quantum theory. (This is explicitly so for the standard model of particle physics and induced from this all fundamental physics ever tested in laboratories; but by all that is known also the remaining ingredient of gravity is fundamentally a quantum theory, see at quantum gravity for comments).
Then I have added at quantum gravity a comment along these line (currently the third paragraph).
Of course much more could and should be said here. But for the moment I urgently need to be doing something else…
60: this is good, now the things are said in the same language so the comparison is apparent. Thanks for putting them in parallel. On the other hand, I am not clear with
The first is supposed to alsways exist for physics to make any sense at all.
I mean one does not expect to have the Lagrangean foliation globally for all Poisson manifolds (or even symplectic!). For example, for almost complex manifolds this would be the complex structure without almost. This is also kind of integrability which, of course, by trivial reasons holds locally, it is not a condition, but I would doubt to have something like that globally, even for some very interesting Poisson manifolds.
@Urs #61: It’s possible that I didn’t make myself clear. I wasn’t doubting that a correct and complete theory of gravity (whatever form it ultimately takes) will be a quantum theory, nor was I commenting on current research programs or whether this or that assumption should be abandoned and other assumptions pursued. I did feel that the sentence quoted in #59 reads as a strong factual declaration, and particularly the assertion that gravity is governed by a quantum theory, however strongly believed, could be construed by some as dogmatic. Thus I felt clarification was in order.
Thanks for responding to this point within the article. With regard to
but by all that is known also the remaining ingredient of gravity is fundamentally a quantum theory
if we couple this with
it is clear that whatver the UV-completion of gravity is, it is out of range of current experiments
a devil’s advocate reaction might be that all you’re really saying is that this “all that is known” is not much in terms of currently testable physics, but at least we have no evidence that quantum gravity is “wrong”, and we (the editorial “we” (-: ) strongly believe in it. Do you think the statement in the article should be interpreted as saying something stronger? (By the way, please treat all “comments” from me about quantum physics as you would student-questions in a class! I understand if you are busy with other tasks.)
a devil’s advocate reaction might be that all you’re really saying is that this “all that is known” is not much in terms of currently testable physics,
I think it’s actually a lot, because it affects the structure of the whole theory, not just a few datatpoints.
There is no consistent way to bring to paper a coupling between a classical theory of gravity and quantum matter using theory that is presently in accord with experiment. This is quite a bit of indirect experimental evidence.
There is no consistent way to bring to paper a coupling between a classical theory of gravity and quantum matter using theory that is presently in accord with experiment.
I’m sorry to be asking naive questions, and I appreciate your patience in answering them.
I’m not sure what the claim is. It sounds like you’re just saying that we have no consistent way of unifying Einstein’s theory with the quantum theory of matter, which I’d be happy to accept on your authority if you say so. (Not that Einstein’s theory, taken on its own, is “wrong” since in fact it passes all experimental tests so far devised. Right?)
The underlying premise seems to be a strong philosophical belief that there is, somewhere out there, a unified theory which covers all physical phenomena (and, as a corollary of sorts, that this will be, with overwhelming likelihood, a quantum theory, and as a second corollary, that Einstein’s theory of gravity will have to be completely revised). But, playing devil’s advocate again, it’s possible that “God” is malicious, and there is no unified theory after all!
I’m not playing games here; I’m trying to get clear on the scope and intended strength of the declaration highlighted in #59 (and revisions thereof), beyond strong philosophical beliefs about the way things are expected to be. Mostly, I want to separate factual assertions from reigning dogma (and I don’t know how else to say it without possibly coming across as harsh – treat it as devil’s advocacy if it helps). Am I making sense to you?
Hey Todd,
good that you are asking. I am being brief since I must not be distracted by this right this moment, for I am supposed to be doing something else. And you can find plenty of in-depth discussion of this in the literature. I’ll dig out page references for you later when I have a second.
But just very briefly:
This is not about philosophy, but about a technical problem. And, yes, the point is precisely that we have two theories which are each confirmed by experiment up to a given level of accuracy: on the one hand Einstein gravity, on the other hand the standard model of particle physics.
Now the technical problem is this: the equations of motion of classical einstein gravity say that “Einstein tensor = Energy-Momentum tensor”. Here into “energy momentum tensor” enters the expression for energy momentum of classical matter fields. But the standard model of particle physics says that these matter fields are not described by classical physics. This is the technical problem: there is no conceivable consistent way to modify Einstein’s equations such as to have a quantum EM tensor on the right. The simple idea of taking the “expectation value” of the quantum matter EM-tensor leads to all kinds of theoretical problems.
It is good that you are pushing this question. This is the big central problem that motivates much of “physics beyond the standard model” since the 1960s. This theoretical mismatch: there is no theory of “classical Einstein gravity coupled to quantum matter.”
In
the point that gravity has to be quantized for consistency is first announced on p. 7, then later discussed in more depth.
In
the issue is introduced right on the first pages.
I’m sure I’ll have more questions on this as we proceed, but thanks!! It’s very kind of you to dig up page references and such, as well as explaining.
For now I’ll just put a pin in that, and we can continue when you have more time. But I gather what you are telling me now is that it’s not just a matter of having two theories which describe different domains of phenomena and we don’t know how to unify them into one, it’s that they can’t both be true! And obviously that’s an extremely serious problem, and not a matter of philosophical belief.
it’s that they can’t both be true!
Certainly neither of them is true in an absolute sense.
As every physical theory so far, both these theories are very accurate in some domain. The point here is is that the idea that classical gravity could remain an accurate description at high energies/small scales is very unlikely, given that gravity couples to stuff and given all the other things that are understood about how stuff behaves at high energies/small scales.
Certainly neither of them is true in an absolute sense.
Oh, I know that. It was just a shortcut way of speaking. :-)
Anyway, I have some reading assignments. :-) I’d love to keep talking about this though.
Re Urs@58, that’s interesting. So maybe all (important) QFTs are quantizations, or duals of quantizations, of action functionals, and there’s a geometric account of this process, which is systematic enough that you can say elsewhere
the (“first”) quantization of $\sigma$-models – at least for the version discussed above – also is a functor.
(You probably say this somewhere else, but it’s from here.)
By the way, how does that idea of Litvinov that took classical mechanics as matrix mechanics over the tropical rig look today? A long time ago we were discussing the states of classical mechanics as modules over the tropical rig.
Also, is there a cohesive homotopy type theoretic account of second quantization?
Thanks, good questions. Right now I am in a haste and can’t reply. Will do so later. But can’t resist to mention a quick reply on your last question:
I have long had the following conjecture:
Recall that we show that natural extended action functionals are morphism $\mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}$ – these are differential coycles. Once we are done (not quite yet) there well be a machine that reads in such a morphism and spits out the corresponding TQFT, which will be a morphism $Bord_n \to \mathcal{C}$.
Conjecture: there is a natural generalization of the previous process that reads in such a directed differential cocycle and spits out something that is then the second quantization of the original morphism.
Notice that this means to take $Bord_n$ as the target space of a $\sigma$-model. Hence a path in target space is now a Feynman diagram and the path integral is now a sum over Feynman diagrams. So that looks right.
It is of course still a good way to go to fill this conjecture with life.
Quick comment…
Each time I manage to find a spare moment (which is rare) to have a look at the page, I immediately get distracted by the “layers”. I think the idea of layers is good and makes sense, but could you consider renaming them?
Maybe something like:
Something more friendly like that?
but could you consider renaming them?
I was thinking about not abbreviating them. Maybe instead of
Layer Mod
Layer Sem
Layer Syn
I should each time write
Model Layer
Semantics Layer
Syntactic Layer
Maybe something like:
- Traditional
- Categorical
- Logical or Homotopical or…
I don’t want to call the model layer “traditional”. There are and will be many things in this layer that are not very “traditional”. For instance in the chapter Differentiation that I am currently writing the discussion of variational caclulus is in the Mod-Layer, but you won’t find many textbooks that set it up the way I do there.
And the reason is a general one, which affects the Mod-Layer throughout: traditional textbooks focus on the notion of smooth manifolds and don’t abandon it even if its inappropriateness flies into their face. Here instead I am doing it right and work with smooth spaces from the beginning. That makes much of the Mod-Layer discussion be very neat, but also non-traditional.
Hi Urs,
I like the non abbreviated version better, and since I brought it up, I could make the revisions. Just let me know of that is what you want to do.
I think it is good to change it. I went to the page thinking,”Finally! An nLab page I should be able to understand.” The scary abbreviations made me retreat mentally (for no good reason I can think of).
Hi Eric, Hi Jim,
thanks to both of you for the feedback. That is useful.
I have now made the changes “Layer Mod” $\mapsto$ “Model Layer”, etc.
I went to the page thinking,”Finally! An nLab page I should be able to understand.”
One word on this: as you already see, my intention is to start each “Model layer” really easy and then gradually become more sophisticated – but also I tend to get distracted from this task, for a wide variety of reasons.
So for this entry and for the $n$Lab in general the following holds: if you run into something that you find incomprehensibly explained or not explained but which you think should have an easy explanation then: ask. If the day had 72 hours I’d spend 24 of them writing expositions, but since it doesn’t I need special incentive to do so. If you ask, I may feel sufficiently pushed to react.
Hi,
Hopefully just one more comment on format and my next will be on content…
The “Contents” are very long. I am generally not a fan of long contents, but I like having the tables expand on the side when you hover over them though. Is it possible to move the long contents into the side panel?
Edit: Rather than be an armchair quarterback, I thought I’d try out something. I created geometryofphysicscontents and included it on the sidebar. I also tried to include a .toc directly in the sidebar. I saw some encouraging results in the Sandbox, e.g. there is a contents in the sidebar containing a link, but do not understand the syntax well enough to get it to work at geometry of physics. Is it even possible to include a .toc in the sidebar?
As I mentioned in another thread, I re-installed the table of contents. Because while a long TOC may be bad, what is worse is a long entry that needs a long TOC but does not have one.
What we need is to have two TOCs: one for only the chapters, one that has also all the subsections.
I had already included such a short high-level TOC, it was in the “Scope”-section. But now I have moved it to the very beginning of the entry.
Does that at least partially address your concern?
I understand. I might have preferred to just add the three lines to create a toc rather than roll it back because there were other minor changes to the sidebar that are now lost. No big deal.
To me, the benefits of the toc are not greater than the distraction of needing to scroll through a super long toc, which is now largely redundant with the table of chapters preceeding it, to get to the contents, but maybe it is just me.
What if we
?
It would be even better if we could nest the hiding feature so that the contents of a chapter appear when you hover over the chapter.
I don’t see why we would want to remove information that some readers may find useful. You don’t need to scroll through the TOC if you already know where you want to go. But for readers who don’t already know what they want to read it is orders of magnitude more convenient to glance through the TOC then glance through the whole entry!
But I suggest we come back to this later when the entry has been developed more. At the moment this whole discussion is a bit premature with the whole entry having only barely started. Let’s see what it looks like later when it is grown up and then spend time with discussing its optimal formatting.
I have kept and keep adding material to geometry of physics, but I have not managed yet to complete a whole new chapter. Rather I ended up adding bits and pieces here and there as I needed them for discussion with students. I thought I’d make a new announcement here only once I have finished a good bit of a new chapter.
But since this state of affairs is not going to change anytime soon, I should just continue to announce the bits and pieces added, anyway.
Today the topic is concretification of smooth groupoids and higher groupoids and how to obtain “moduli stacks of connections on some $X$” this way:
This is a generalization of the earlier discussion of Smooth moduli spaces of differential forms, where the key is that instead of just using the ordinary image of the dissolving map $X \to \sharp X$ into the sharp modality to concretify, we need all the n-images of this map, and combine them degreewise.
Might that confuse a reader to have in ’3. term introduction rule’ that an element of an object $A$ is a morphism from any object $Q$, whereas just below in ’Dictionary: type theory / category theory’, an element corresponding to a term $a$ of $A$ is a morphism only from $\ast$?
I see what you mean. So further below in that “Dictionary” entry is the more general case of terms on context. There the “$X$” is essentially (not quite exactly, actually) what used to be the $Q$ before. Do you think it would be better to call them both $X$ are both $Q$?
I’m not sure that matters so much about $X$ or $Q$, but it seems to me that someone coming from a set theoretic background, who has just got the idea that a set theoretic element, $a$, is a map from a singleton to the set (and set element is the first thing they see at element) and then links this to a term $a:A$, will be confused about the use of a different domain.
Urs, now that you’re getting back to “Geometry of Physics”, I took the opportunity of incorporating your words in #67 (which I found very helpful) into the Idea section of quantum gravity. I hope you don’t mind. I changed a word here and there just for the sake of the English.
Thanks, Todd! Much appreciated.
I added some more hyperlinks, for instance to Einstein tensor, energy-momentum tensor, Einstein equations and… (in production) field (physics)…
There is now a good bit of material in a new section
It’s still a bit rough here and there, but should be a start. Mostly this reproduces the nLab entry field (physics), for the moment.
Hm, turns out I have section headlines of depth 7 now and they are no longer correctly displayed. Need to think about what to do. Is this something one could fix with CSS?
I am writing a section to lead in to the Physics-chapter at geometry of physics. It is called
There are currently 7 subsections to that. The first five should be readable, the sixth is still a bit telegraphic and sketchy, the seventh for the time being just some keywords. Will continue to tomorrow.
The same material is currently also at extended Lagrangian. That page has the advantage that it loads orders of magnitude faster than geometry of physics.
Today I gave the first of three lecture sessions at
Workshop on Topological Aspects of Quantum Field Theories
Singapore
(14 - 18 Jan 2013)
on infinity-Chern-Simons theory (schreiber). This first session followed the section
During the talk I noticed of course a few more typos, which I have fixed now. There will be more typos left, and more fine-tuning wouldn't hurt either. If you point me to anything that bugs you, thwn I'll try to react and improve on the text further.
I corrected a few typos in extended Lagrangian, but haven’t copied these over to geometry of physics.
Thanks a whole lot! That's nice.
I have copied the whole chunk back to geometry of physics now. Thanks again.
Today I gave the second of three lecture sessions at
Workshop on Topological Aspects of Quantum Field Theories
Singapore
(14 - 18 Jan 2013)
on infinity-Chern-Simons theory (schreiber). This second session followed the section
During the talk I noticed of course a few more typos, which I have fixed now. There will be more typos left, and more fine-tuning wouldn’t hurt either. If you point me to anything that bugs you, then I’ll try to react and improve on the text further.
there is now the beginning of a new chapter:
(for the moment in a separate entry, for ease of editing). So far this mainly consists of a subsection on 2-modules.
Does geometry of physics exist? I get only ths:
Application error (Apache)
Something very bad just happened. I just know it. Do you smell smoke?
Yeah, in principle it exists. The page did display until, it seems, the nLab software was updated a while back.
Probably I should try to take the page apart into smaller subpages. I need to see if I can find time for this and how much fiddling it will take.
Or I just wait and pray for the bug to be ironed out.
I have now taken out a bunch of chapters from geometry of physics and put them into separate entries. For instance
and a few more.
I still get that smoke error every now and then here and there, but reloading often helps. Please try reloading geometry of physics to see if it displays now.
Of course now all the cross-links involving the sections that I took out are broken, as are all the pointers to the references; the main table of contents is now incoherent (since it doesn’t see the subsections of the chapters taken out anymore) etc., But I don’t have the time now to fix all this.
It’s just too bad that the nLab software cannot handle this.
It displays without smoke for me.